[Proof of Thm-local]
For the problem equ-phizeta, ic-pertur, it is standard (see [1]} for instance) to prove the local existence in time of the solution \( (\phi ,\zeta ) \) , which is periodic in the transverse directions \( x^{\perp }\in \mathbb {T}^2 \)  and satisfies that \( (\phi , \zeta ) \in C(0,T_0;H^3(\Omega )) \)  and \( \nabla \zeta \in L^2(0,T_0; H^3(\Omega )) \)  for some constant \( T_0>0. \)
